Having always been attracted by pure mathematics and philosophical questions, I've indulged this in trying to construct an alternative modern foundation for differential calculus. It draws on our intuitions of motion, and resurrects Isaac Newton's theory of fluents and fluxions.
The following monograph sets out the gory details. It's written quite formally in the general style of discussion, definition, theorem and proof:
Revisiting Fluxions (opens in a new window or tab as a PDF file).
The aim is to add to existing modern approaches by exhibiting a method that draws on our kinematic intuitions and is logically rigorous. It's based on what it might mean for one object to be going faster than another. The key intuition is one which is very easy to grasp: that the faster you go, the further you travel in a given time.
I've tried to show how a particular treatment of fluents and fluxions can be used to develop a number of fluent-based precursors to differentiation, ultimately providing a definition of the derivative of a function very similar to Newton's formulation as a ratio of two fluxions.
The approach purposely contrasts with three historic starting points in deriving differential calculus:
Each of these can be viewed from the dual perspective of intuition and rigour. For example geometric arguments appeal to intuition but can fall short of modern standards of rigour. The arithmetisation of analysis provided a ground-breaking rigorous foundation, which for some can seem overly abstracted and remote from its original aims, coming at the expense of natural intuitiveness.
Although some of my own personal biases may be evident, the approach I've set out using fluents and fluxions isn't a polemic. Instead, it explores the consequences of redefining these terms using concepts that weren't available in Newton's day, then applying them to a formal statement of what it means for one varying quantity to be advancing faster than another.
Some of the key results demand more work than I would have liked. Nevertheless, the result for me has seemed to be unexpectedly rich and thought-provoking, suggesting that there really may be a viable fourth starting point for calculus, based on intuitions of time in continuous progression and values in a dynamic state of flow.